Name: Optical Instruments: Spectrometer
Uploaded: 2019-07-19
Duration: 7 min 33 s
Description: The basic function of a spectrometer is to take in light, break it into its spectral components, digitise the signal as a function of wavelength, and read it...

Question

Derive the expression for focal length of a spherical mirror

F = \frac{R}{2} .

Accepted Answer

Let us consider a concave mirror such that its radius of curvature is very much larger than the diameter of its aperture.

Suppose a parallel ray of light is incident on the mirror as shown in the figure. Let this incident ray make an angle $θ$ with normal to the surface of the mirror. Since this mirror is part of a circle, the normal drawn to the surface of the circle passes through the centre $(C)$ of the mirror. We know that rays of light parallel to the principal axis passing through the focus $(F)$ of a concave mirror, after reflection.

According to the law of reflections, the angle of incidence and angle of reflection are equal. Therefore, $∠ BAC = ∠ FAC = θ$ as shown in the figure. Since $BA$ and $PC$ are parallel, $∠ BAC = ∠ ACF = θ$ .

Therefore, from exterior angle theorem $∠ AFN = ∠ ACF + ∠ CAF = 2 θ$ .

Now drop a normal $CP$ from point $A$ . Let the foot of this normal be $N$ .

Here, $\tan θ = \frac{AN}{CN} \Rightarrow AN = CN \tan θ . . . . . . (i)$

Now, $\tan 2 θ = \frac{AN}{FN} \Rightarrow AN = FN \tan 2 θ . . . . . (ii)$ . From equation $(i)$ and $(ii)$ we get,

$CN \tan θ = FN \tan 2 θ \Rightarrow \frac{CN}{FN} = \frac{\tan 2 θ}{\tan θ} . . . . . . (iii)$ . Since the radius of curvature is very much larger than the diameter of its aperture, $NP$ is very small compared to $CN$ and $CP$ and $θ$ will be a small angle. Therefore, $CN \approx CP$ and $FN \approx FP$ . For small angles $\tan θ = θ$ and $\tan 2 θ = 2 θ$ . Therefore, equation $(iii)$ can be written as

$\Rightarrow \frac{CP}{FP} = \frac{2 θ}{θ} \Rightarrow FP = \frac{CP}{2}$ , and $CP = R$ and $FP = f$ . Hence, $f = \frac{R}{2}$ .

Note that this relation between the radius of curvature $(R)$ of a concave mirror and the focal length $(f)$ of the mirror, which is $f = \frac{R}{2}$ , is true only when the $R$ is very much larger than the diameter of its aperture.

Derive the expression for focal length of a spherical mirror $F = \frac{R}{2} .$

Important Questions on Ray Optics and Optical Instruments

Learn and Prepare for any exam you want !

Derive the expression for focal length of a spherical mirror F=R2.

Important Questions on Ray Optics and Optical Instruments

Learn and Prepare for any exam you want !

Derive the expression for focal length of a spherical mirror $F = \frac{R}{2} .$